# 2017-18

*(Covering topics related to random matrices, representation theory, integrable systems and interacting stochastic particle systems)*

**Seminars are held on Tuesdays at 12:00, B3.02**

## Term 1

3rd October. **Roger Tribe** (Warwick).* A short non-solution to the KPZ problem*.

**Abstract**. I will go through a very short formal derivation of the Tracy-Widom distribution for the fluctuations of the KPZ interface and descirbe the remaining gap between the formal steps and the rigorous proof.

10th October. **Will FitzGerald** (Warwick). *Reflected Brownian motions and random matrix theory.*

**Abstract**. Systems of Brownian motions with one-sided reflection in the KPZ universality class are described by distributions from random matrix theory. We will show how this connection can be obtained directly from Schutz-type transition probabilities. This can be extended in the presence of a wall, drifts and for particle systems connected to last passage percolation.

**References. **

Kurt Johansson, *A multi-dimensional Markov chain and the Meixner ensemble*

https://projecteuclid.org/euclid.afm/1485907101

Jon Warren, *Dyson's Brownian motions, intertwining and interlacing*

https://projecteuclid.org/euclid.ejp/1464818490

Alexei Borodin, Patrik Ferrari, Michael Prahofer, Tomohiro Sasamoto and Jon Warren, *Maximum of Dyson Brownian motion and non-colliding systems with a boundary*

https://projecteuclid.org/euclid.ecp/1465234756

17th October **Will FitzGerald** (Warwick). Reflected Brownian motions and random matrix theory-II.

24th October. **Theodoros Assiotis** (Warwick). *Matrix Bougerol identity and the Hua-Pickrell measures.*

**Abstract**: I will talk about how one can extend to the matrix setting two celebrated one-dimensional identities of Bougerol and Dufresne related to exponential functionals of Brownian motion.

31st October. **Elia Bisi** (Warwick). *Point-to-line last passage percolation via symplectic Schur functions*.

**Abstract.** We discuss a new formula, in terms of symplectic Schur functions, for the point-to-line last passage percolation with exponentially distributed waiting times. We then show how to derive, in the scaling limit, Sasamoto’s Fredholm determinant formula for the GOE Tracy-Widom distribution. If time permits, we also go through the last passage percolation model in the point-to-half-line geometry, where the asymptotic distribution is instead the marginal of the $Airy_{2 \to 1}$ process.

7th November. **Elia Bisi** (Warwick). *Point-to-line last passage percolation via symplectic Schur functions-II. **(Point-to-line polymers and orthogonal Whittaker functions.)*

**Abstract.** We study a one dimensional directed polymer model in an inverse-gamma random environment, known as the log-gamma polymer, in the point-to-line geometry. Via the use of A.N.Kirillov’s geometric Robinson-Schensted-Knuth correspondence, we compute the Laplace transform of the partition function as an integral of orthogonal Whittaker functions. In the zero temperature limit, we recover the formula we discussed in the previous talk for the distribution of the point-to-line last passage percolation with exponentially distributed waiting times.

14th November. No seminar

21st November. **Theodoros Assiotis** (Warwick). *Determinantal structures in (2+1)-dimensional growth and decay models.*

**Abstract**. I will talk about an inhomogeneous growth and decay model with a wall present in which the growth and decay rates on a single horizontal slice of the surface can be chosen essentially arbitrarily depending on the position. This model turns out to have a determinantal structure and most remarkably for a certain, the fully packed, initial condition the correlation kernel can be calculated explicitly in terms of one dimensional orthogonal polynomials on the positive half line and their orthogonality measures.

28th November. ** Nick Simm** (Warwick). *The real spectrum for products of non-Hermitian random matrices.*

**Abstract**: Let $M$ be a matrix whose entries are i.i.d. standard Gaussian variables. A result of Edelman, Kostlan and Shub says that the expected number of real eigenvalues of $M$ grows like $\sqrt{2N/\pi}$ as the size $N$ of the matrix grows. I will discuss how this estimate should be modified for products of such random matrices, leading one to conclude that, on average, multiplication enhances the number of real eigenvalues. I will also present a result describing the asymptotic density of eigenvalues on the real line, proving a conjecture of Peter Forrester and Jesper Ipsen. The results are based on the asymptotic analysis of certain special functions known as “Meijer G”.

5th December. Discussion of the reading list for the next term

## Term 2

9th January. *No seminar*

16th January. **Jon Warren** (Warwick). *A first look at the Okounkov-Vershik approach to representations of **symmetric groups.*

**Abstract.** The talk will be an overview of the paper: Vershik, Anatoly M., and A. Yu Okounkov. "A new approach to the representation theory of the symmetric groups. II." Journal of Mathematical Sciences 131.2 (2005): 5471-5494. Hopefully it will be accessible to people with little or no knowledge of representation theory- I will give some very simple examples to illustrate the results in the paper.

23th January. **Bruce Westbury** (Warwick). *A second look at the Okounkov-Vershik approach to representations of symmetric groups.*

30th Janauary. **Bruce Westbury** (Warwick). *A final look at the Okounkov-Vershik approach to representations of symmetric groups*.

6th February.** Roger Tribe** (Warwick). *The Brownian Web and the Brownian Net. *

**Abstract.** This Brownian Web is a random set of coalescing 1-d Brownian paths ‘starting from every space-time point’.

It was constructed by Toth and Werner (as a tool for studying non-intersecting walks), but has been

an object of much recent exploration. The survey [SSS] describes the current state of the art and

has a nice list of open problems. The Web is a rich enough object to contain all the continuum integrable

point processes that Oleg and I have been studying with our students. I want to give an introduction,

explain briefly how to find the integrable structures, and my target is to go through the construction of some

interesting subsets (for example the Brownian Net which describes branching coalescing particles).

13th February. **Roger Tribe** (Warwick). *The Brownian Web and the Brownian Net-II.*

20th February. **Oleg Zaboronski** (Warwick). *Coalescing Brownian motions and the Web. *

**Abstract**. I will show how to apply the Web properties discussed by Roger to the derivation of the Pfaffian structure for the Arratia flow.

27th February. *No seminar*

6th March. **Roger Tribe** (Warwick). *The Brownian Web and the Brownian Net-III.*

13th March. **Sakis Tsareas** (Warwick).* Conditioning of determinantal and pfaffian point processes*

**Abstract:** I'll go over two identities, a determinantal and a pfaffian one, and show how they can be used to attain

explicit formulas for the kernel of the new point processes after conditioning on a set of points.

In the end I'll focus on the continuous determinantal point processes and show how these results can potentially be used to

prove rigidity related theorems.

Reading list for Term 2

1. Okounkov2003

3. SSS review of the Brownian Net

## Term 3

05th June. **Will FitzGerald** (Warwick). *Kac polynomials and random analytic functions*

**Abstract**:The zero sets of random polynomials and random analytic functions with Gaussian coefficients form interesting point processes with repulsive interactions. We consider a special case where the zero set is a Pfaffian point process.

**References:**

[1] **Hough, Krishnapur, Peres and Virag. ***Zeroes of Gaussian analytic functions and determinantal point processes*

[2] **Matsumoto and Shirai**. *Correlation functions for zeroes of a Gaussian power series and Pfaffians*